yes, proof by counter-example is obvious here. i was hoping for something else. because this seems to imply a larger problem, namely, the distribution of powers over a product. if it doesn't work for the 1/2 power, that would mean it probably won't work for all or some other powers, right? a proof i would like to see is when this move is illegal, or is it always illegal? is false for ? etc

Let be defined as . Then is not a well defined map, because there exists such that maps to two different elements in

This doesn't really apply to your proof, but in general be careful when you use the square root with complex numbers, as you'll often get multiple answers. In fact, when you use the square root with real numbers, there are multiple possible answers, but it's convention to ignore the negative parts. I could do the same proof without complex numbers.

We know that and the square root and square operations are inverses so they cancel and .

And we know that .
example:

So .

So . gg.

Where is the problem with this?

Originally Posted by mnova

[I hope I did the LaTex correctly.]

We know that and the square root and square operations are inverses so they cancel and .

Let be defined as . Then is not a well defined map, because there exists such that maps to two different elements in

Yes it is well-defined. There are two ways to define the square root, but the one used in post #1 is the principal square root: . Once you settle with this definition the square root function is well-defined.